Novel tomography techniques and parameter identification problems

Transcription

1 Novel tomography techniques and parameter identification problems Bastian von Harrach Department of Mathematics - M1, Technische Universität München Colloquium of the Institute of Biomathematics and Biometry, Helmholtz Zentrum München, 23 June 2010.

2 Overview Parameter identification problems in diffuse optical tomography (DOT) Linearized reconstruction algorithms in electrical impedance tomography (EIT)

3 Parameter identification problems in DOT

4 Diffuse optical tomography Diffuse optical tomography (DOT): Transilluminate biological tissue with visible or near-infrared light Goal: Reconstruct spatial image of interior physical properties. Relevant quantities (in diffusive regime): Scattering Absorption Applications: Breast cancer detection Bedside-imaging of neonatal brain function Topical reviews: Arridge & Schotland (2009), Gibson, Hebden & Arridge (2005), Arridge (1999)

5 Mathematical Model General Forward Model: Photon transport models (Boltzmann transport equation) Recent review: Bal, Inverse Problems 25, (48pp), For highly scattering media: DC diffusion approximation for photon density u: (a u)+cu = 0 in B R n, u : B R: photon density a : B R: diffusion/scattering coefficient c : B R: absorption coefficient Boundary measurements (idealized): Neumann and Dirichlet data u S, a ν u S on S B. Remaining boundary assumed to be insulated, a ν u B\S = 0.

6 Forward & inverse problem DC diffuse optical tomography: (a u)+cu = 0 in B R n, n 2, B bounded with smooth boundary, S B open part, a,c L +(B). { g L 2 (S)! solution u H 1 g on S, (B) : a ν u B = 0 on B \S. (Local) Neumann-to-Dirichlet map Λ a,c : g u S, L 2 (S) L 2 (S) is linear, compact and self-adjoint. Inverse Problem: Can we reconstruct a and c from Λ a,c?

7 Non-uniqueness DC diffuse optical tomography: (a u)+cu = 0 Arridge/Lionheart (1998 Opt. Lett ): v := au solves v +ηv = 0, with η = a a + c a. a = 1 around S (u S,a ν u S ) = (v S, ν v S ). Λ a,c only depends on effective absorption η = η(a,c). Absorption and scattering effects cannot be distinguished. (Note: Argument requires smooth scattering coefficient a).

8 Experimental results Theory: Absorption and scattering effects cannot be distinguished. Practice: Successful separate reconstructions of absorption and scattering (from phantom experiment using dc diffusion model!) Pei et al. (2001), Jiang et al. (2002), Schmitz et al. (2002), Xu et al. (2002) Pei et al. (2001): Practice contradicts theory! As a matter of established methodological principle (...) empirical facts have the right-of-way; if a theoretical derivation yields a conclusion that is at odds with experimental results, the reconciliatory burden falls on the theorist, not on the experimentalist.

9 New uniqueness result Theorem (H., Inverse Problems 2009) a 1,a 2 L +(B) piecewise constant c 1,c 2 L +(B) piecewise analytic If Λ a1,c 1 = Λ a2,c 2 then a 1 = a 2 and c 1 = c 2. Piecewise constantness seems fulfilled for phantom experiments. Result reconciles theory with practice. Measurements contain more than just the effective absorption! Next slides: Idea of the proof using monotony and localized potentials.

10 Monotony result Lemma Let a 1,a 2,c 1,c 2 L +(B). Then for all g L 2 (S), ( (a2 a 1 ) u 1 2 +(c 2 c 1 ) u 1 2) dx B (Λ a1,c 1 Λ a2,c 2 )g,g ( (a2 a 1 ) u 2 2 +(c 2 c 1 ) u 2 2) dx, B u 1,u 2 H 1 (B): solutions for (a 1,c 1 ), resp., (a 2,c 2 ). Can we control u j 2 and u j 2?

11 Virtual Measurements Ω L Ω S L Ω : (H 1 (Ω)) L 2 (S), f u S, where u H 1 (B) solves (a u v +cuv) dx = f,v B (essentially: (a u)+cu = f)

12 Virtual Measurements Unique continuation: If Ω 1 Ω 2 =, B\(Ω 1 Ω 2 ) connected neighbourhood of S then R(L Ω1 ) R(L Ω2 ) = 0. Dual operator: L Ω : L2 (S) H 1 (Ω), g u Ω, where u solves { g on S, (a u)+cu = 0, a ν u B = 0 else.

13 Some functional analysis Lemma Let X,Y be two reflexive Banach spaces, A L(X,Y), y Y. Then y R(A) iff y,y C A y y Y. Corollary If L Ω 1 g C L Ω 2 g fluxes g, i.e., if u Ω1 H 1 C u Ω2 H 1 for the corresponding densities u, then R(L Ω1 ) R(L Ω2 ). Contraposition R(L Ω1 ) R(L Ω2 ) (g k ) such that the solutions (u k ) satisfy u k Ω1 H 1 (Ω 1 ) and u k Ω2 H 1 (Ω 2 ) 0.

14 Some functional analysis Similarly (by unique continuation and regularity) L(L 2 (Ω)) L(H 1 (Ω) ). (g k ) such that the solutions (u k ) satisfy u k Ω H 1 (Ω) and u k Ω L 2 (Ω) 0.

15 Localized potentials Lemma There exist solutions u with B B u H 1 (B\O) small B S O B { u H 1 (O) large u L 2 (O) small u H 1 (B\O) small O S O u L 2 (O ) large

16 Localized potentials Lemma There exist solutions u with Ω B B u H 1 (B\O Ω) small { u H 1 (Ω) large u L 2 (Ω) small O S B Ω Ω B u H1(B\O Ω) small u L 2 (Ω ) large S O

17 Proof of uniqueness Monotony ( (a2 a 1 ) u 1 2 +(c 2 c 1 ) u 1 2) dx B (Λ a1,c 1 Λ a2,c 2 )g,g ( (a2 a 1 ) u 2 2 +(c 2 c 1 ) u 2 2) dx, B Proof of the uniqueness result (very sketchy...) Start with region next to S Use loc. pot. with u 2 in that region a 1 = a 2 Then use loc. pot. with u 2 in that region c 1 = c 2 Repeat over all regions.

18 Uniqueness or not? Arridge/Lionheart (1998): Non-uniqueness for general smooth (a, c). H. (2009): Uniqueness for piecew. constant a, piecew. analytic c. What information about (a,c) does Λ a,c really contain? Formally(!), Λ a,c can only determine η = a a + c a. Jumps in a or a distributional singularities in a. Bold guess: Maybe Λ a,c determines η where a and c are smooth, jumps in a and a. (However, note that a/ a is not well-defined for non-smooth a...)

19 Exact characterization Theorem ( H., submitted for publication) Let a 1,a 2,c 1,c 2 L +(B) piecewise analytic on joint partition B = O 1... O J Γ, O 1... O J = B Γ. Then, Λ a1,c 1 = Λ a2,c 2 if and only if (a) a 1 S = a 2 S, and ν a 1 S = ν a 2 S on S, (b) ν a 1 a B\S = νa 2 1 a B\S 2 a2 (c) η 1 := a 1 + c 1 = + c 2 =: η 2 a1 a 1 a2 a 2 on B \S, on B \Γ, (d) a 1 + Γ a1 Γ = a+ 2 Γ a2, and Γ [ ν a 2 ] Γ a 2 Γ = [ νa 1 ] Γ a 1 Γ on Γ.

20 Proof Proof relies on more general monotony results, e.g., g (Λ a2,c 2 Λ a1,c 1 )g ds S (η 1 η 2 )a 2 u 2 2 dx B\Γ + + ( ( a2 ν a 1 1 )gu 2 ds νa 2 a1 B 2a 1 2a 2 { ( [ ] ) 1 a2 [ ν a 2 ] 2 Γ ν a 1 u a 1 S Γ Γ ) a 2 u 2 2 ds [ ] a2 a1 Γ a 1 ν u 1 u 2 } ds Then localized potentials are used to control u H 1, u L 2 on subsets and u Σ L 2 on boundary parts.

21 Linearized reconstruction algorithms in EIT

22 EIT Electrical impedance tomography (EIT): Apply currents σ ν u B (Neumann boundary data) Electric potential u solves (σ u) = 0 in B Measure voltages u B (Dirichlet boundary data) Current-Voltage-Measurements Neumann-to-Dirichlet map Λ(σ)

23 Inverse problem Non-linear forward operator of EIT Λ : σ Λ(σ), L +(B) L(L 2 ( B)) Inverse problem of EIT: Λ(σ) σ? Localized potentials Uniqueness for piecewise analytic conductivities already known: Druskin ( ), Kohn/Vogelius ( )

24 Linearization Generic approach: Linearization Λ(σ) Λ(σ 0 ) Λ (σ 0 )(σ σ 0 ) σ 0 : known reference conductivity / initial guess /... Λ (σ 0 ): Fréchet-Derivative / sensitivity matrix. Λ (σ 0 ) : L +(B) L(L 2 ( B)). Solve linearized equation for difference σ σ 0. Often: supp(σ σ 0 ) B compact. ( shape / inclusion )

25 Linearization Linear reconstruction method e.g. NOSER (Cheney et al., 1990), GREIT (Adler et al., 2009) Solve Λ (σ 0 )κ Λ(σ) Λ(σ 0 ), then κ σ σ 0. Multiple possibilities to measure residual norm and to regularize. No rigorous theory for single linearization step. Almost no theory for Newton iteration: Dobson (1992): (Local) convergence for regularized EIT equation. Lechleiter/Rieder(2008): (Local) convergence for discretized setting. No (local) convergence theory for non-discretized case!

26 Linearization Linear reconstruction method e.g. NOSER (Cheney et al., 1990), GREIT (Adler et al., 2009) Solve Λ (σ 0 )κ Λ(σ) Λ(σ 0 ), then κ σ σ 0. Seemingly, no rigorous results possible for single linearization step. Seemingly, only justifiable for small σ σ 0 (local results). Here: Rigorous and global(!) result about the linearization error.

27 Exact Linearization Theorem (H./Seo, accepted to SIAM J. Math. Anal.) Let κ, σ, σ 0 piecewise analytic and Λ (σ 0 )κ = Λ(σ) Λ(σ 0 ). Then (a) supp B κ = supp B (σ σ 0 ). (b) σ 0 σ (σ σ 0) κ σ σ 0 on the bndry of supp B (σ σ 0 ). supp B : outer support ( = support, if support is compact and has conn. complement) Exact solution of lin. equation yields correct (outer) shape. No assumptions on σ σ 0! Linearization error does not lead to shape errors. Proof: Combination of monotony and localized potentials.

28 Proof Exact linearization: Λ (σ 0 )κ = Λ(σ) Λ(σ 0 ) Monotony: For all reference solutions u 0 : (σ σ 0 ) u 0 2 dx B σ 0 g,(λ(σ) Λ(σ 0 ))g } {{} B σ (σ σ 0) u 0 2 dx. = κ u 0 2 dx B Use localized potentials to control u 0 2 supp Ω κ = supp Ω (σ σ 0 )

29 Non-exact Linearization? Theorem requires Λ (σ 0 )κ = Λ(σ) Λ(σ 0 ). Existence of exact solution is unknown! In practice: finite-dimensional, noisy measurements Proof only requires monotony estimate. Approximately solve linearized equ. s.t. estimate is still fulfilled Globally convergent algorithm for non-exact linearization (not nicely implementable!) Open questions / ongoing work Obtain convergent algorithm in nice form (e.g., Tikhonov regularization with special penalty term)

30 Experimental result Similar results hold if reference measurement is taken at the same object but with another frequency ( complex conductivites, weighted frequency differences) Next slide: Experimental tests of a heuristic combination of linearization and localized potentials.

31 Experimental result (H./Seo/Woo, to appear in IEEE Trans. Med. Imaging)

32 Conclusion Novel tomography techniques lead to mathematical parameter identification problems. Uniqueness questions may have non-trivial answers. In diffuse optical tomography: Uniqueness for piecewise constant coefficients (even for piecw. linear/analytic) No uniqueness for general smooth coefficients Close interplay between uniqueness arguments and convergent reconstruction algorithms.